5.3-HYPERBOLA-01-THEORY

5.3.1 Definition. A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity. Fixed point is called focus, fixed straight line is called directrix and the constant ratio is called eccentricity of the hyperbola. Eccentricity is denoted by e and e > 1. A hyperbola is the particular case of the conic When , i.e., and . Let is the focus, directrix is the line and the eccentricity is e. Let be a point which moves such that   Hence, locus of is given by Which is a second degree equation to represent a hyperbola (e > 1). Example: 1 The equation of the conic with focus at (1, – 1), directrix along and with eccentricity is (a) (b) (c) (d) Solution: (c) Here, focus (S) = (1, –1), eccentricity (e)= From definition ,  =  , which is the required equation of conic (Rectangular hyperbola) Example: 2 The centre of the hyperbola is (a) (2, 3) (b) (– 2, – 3) (c) (–2, 3) (d) (2, – 3) Solution: (a) Here , , , Centre of hyperbola = = = (2, 3) 5.3.2 Standard equation of the Hyperbola . Let S be the focus, ZM be the directrix and e be the eccentricity of the hyperbola, then by definition,      , where This is the standard equation of the hyperbola. Some terms related to hyperbola : Let the equation of hyperbola is (1) Centre : All chords passing through C are bisected at C. Here (2) Vertex: The point A and A where the curve meets the line joining the foci S and S  are called vertices of hyperbola. The co-ordinates of A and A are (a, 0) and (– a, 0) respectively. (3) Transverse and conjugate axes : The straight line joining the vertices A and A is called transverse axis of the hyperbola. The straight line perpendicular to the transverse axis and passing through the centre is called conjugate axis. Here, transverse axis = Conjugate axis = (4) Eccentricity : For the hyperbola We have , (5) Double ordinates : If Q be a point on the hyperbola, QN perpendicular to the axis of the hyperbola and produced to meet the curve again at . Then is called a double ordinate at Q. If abscissa of Q is h, then co-ordinates of Q and are and respectively. (6) Latus-rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axis is called latus-rectum. Length of latus-rectum and end points of latus-rectum ; ; respectively. (7) Foci and directrices: The points and are the foci of the hyperbola and ZM and are two directrices of the hyperbola and their equations are and respectively. Distance between foci and distance between directrices . (8) Focal chord : A chord of the hyperbola passing through its focus is called a focal chord. (9) Focal distance : The difference of any point on the hyperbola from the focus is called the focal distance of the point. From the figure, , = The difference of the focal distance of a point on the hyperbola is constant and is equal to the length of transverse axis. Transverse axis Example: 3 The eccentricity of the hyperbola which passes through (3, 0) and is (a) (b) (c) (d) None of these Solution: (b) Let equation of hyperbola is . Point (3, 0) lies on hyperbola So, or or and point also lies on hyperbola. So, Put we get, or or or or We know that . Putting values of and or or or . Example: 4 The foci of the hyperbola are (a) (b) (c) (d) Solution: (c) The equation of hyperbola is . Now,   . Hence foci are = i.e., Example: 5 If the foci of the ellipse = 1 and the hyperbola coincide, then the value of is (a) 1 (b) 5 (c) 7 (d) 9 Solution: (c) For hyperbola, Therefore foci = . Therefore foci of ellipse i.e., (For ellipse )  Hence . Example: 6 If PQ is a double ordinate of hyperbola such that CPQ is an equilateral triangle, C being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies [EAMCET 1999] (a) (b) (c) (d) Solution: (d) Let ; be end points of double ordinates and is the centre of the hyperbola Now ; Since ,          5.3.3 Conjugate Hyperbola . The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called conjugate hyperbola of the given hyperbola. Hyperbola Fundamentals or Centre (0, 0) (0, 0) Length of transverse axis 2a 2b Length of conjugate axis 2b 2a Foci Equation of directrices Eccentricity Length of latus rectum Parametric co-ordinates , Focal radii & & Difference of focal radii 2a 2b Tangents at the vertices Equation of the transverse axis Equation of the conjugate axis Note :  If e and are the eccentricities of a hyperbola and its conjugate, then .  The foci of a hyperbola and its conjugate are concyclic. Example: 7 The eccentricity of the conjugate hyperbola of the hyperbola , is (a) 2 (b) (c) 4 (d) Solution: (a) The given hyperbola is . Here and Since    If is the eccentricity of the conjugate hyperbola, then = 1   . 5.3.4 Special form of Hyperbola . If the centre of hyperbola is (h, k) and axes are parallel to the co-ordinate axes, then its equation is . By shifting the origin at (h, k) without rotating the co-ordinate axes, the above equation reduces to , where . Example: 8 The equation of the hyperbola whose foci are (6, 4) and (– 4, 4) and eccentricity 2 is given by [MP PET 1993] (a) (b) (c) (d) Solution: (a) Foci are (6, 4) and (– 4, 4) and . Centre is So,   and Hence, the required equation is or Example: 9 The equations of the directrices of the conic are (a) (b) (c) (d) Solution: (c)  Equation of directrices of are Here , . Hence,  . 5.3.5 Auxiliary circle of Hyperbola . Let be the hyperbola with centre C and transverse axis . Therefore circle drawn with centre C and segment as a diameter is called auxiliary circle of the hyperbola Equation of the auxiliary circle is Let Here P and Q are the corresponding points on the hyperbola and the auxiliary circle (1) Parametric equations of hyperbola : The equations and are known as the parametric equations of the hyperbola . This ( ) lies on the hyperbola for all values of . Position of points Q on auxiliary circle and the corresponding point P which describes the hyperbola and 0   2  varies from 0 to I I to II III to III II to IV IV Note :  The equations and are also known as the parametric equations of the hyperbola and the co-ordinates of any point on the hyperbola are expressible as where and . Example: 10 The distance between the directrices of the hyperbola , is (a) (b) (c) (d) Solution: (c) Equation of hyperbola is  ,  Here . Now Distance between directrices = = . 5.3.6 Position of a point with respect to a Hyperbola . Let the hyperbola be . Then will lie inside, on or outside the hyperbola according as is positive, zero or negative. Example: 11 The number of tangents to the hyperbola through (4, 1) is [AMU 1998] (a) 1 (b) 2 (c) 0 (d) 3 Solution: (c) Since the point (4, 1) lies inside the hyperbola ; Number of tangents through (4, 1) is 0. 5.3.7 Intersection of a Line and a Hyperbola. The straight line will cut the hyperbola in two points may be real, coincident or imaginary according as . Condition of tangency : If straight line touches the hyperbola , then . 5.3.8 Equations of Tangent in Different forms. (1) Point form : The equation of the tangent to the hyperbola at is . (2) Parametric form : The equation of tangent to the hyperbola at is (3) Slope form : The equations of tangents of slope m to the hyperbola are and the co-ordinates of points of contacts are . Note :  If the straight line touches the hyperbola , then .  If the straight line touches the hyperbola , then  Two tangents can be drawn from an outside point to a hyperbola. Important Tips  For hyperbola and , the equation of common tangent is , points of contacts are and length of common tangent is .  If the line touches the hyperbola at the point , then . Example: 12 The value of m for which is a tangent to the hyperbola , is [Karnataka CET 1993] (a) (b) (c) (d) Solution: (a) For condition of tangency, . Here , Then,    Example: 13 If and are the slopes of the tangents to the hyperbola which pass through the point (6, 2), then (a) (b) (c) (d) Solution: (a, b) The line through (6, 2) is  Now, from condition of tangency   Obviously, its roots are and , therefore and Example: 14 The points of contact of the line with is (a) (4, 3) (b) (3, 4) (c) (4, – 3) (d) None of these Solution: (a) The equation of line and hyperbola are .....(i) and .....(ii) From (i) and (ii), we get  or  From (i), so points of contact is (4, 3) Trick : Points of contact are . Here , and . So the required points of contact is (4, 3). Example: 15 P is a point on the hyperbola is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to (a) (b) (c) (d) Solution: (b) Let be a point on the hyperbola. Then the co-ordinates of N are . The equation of the tangent at is This meets x-axis at ; Example: 16 If the tangent at the point on the hyperbola is parallel to , then the value of is [MP PET 1998] (a) (b) (c) (d) Solution: (c) Here and Differentiating w.r.t.  and Gradient of tangent ; .....(i) But tangent is parallel to ; Gradient ......(ii) From (i) and (ii),  , Example: 17 The slopes of the common tangents to the hyperbola and are [Roorkee 1997] (a) – 2, 2 (b) – 1, 1 (c) 1, 2 (d) 2, 1 Solution: (b) Given hyperbola are .....(i) and .....(ii) Any tangent to (i) having slope m is .....(iii) Putting in (ii), we get,   .....(iv) If (iii) is a tangent to (ii), then the roots of (iv) are real and equal. Discriminant = 0; = =      5.3.9 Equation of Pair of Tangents. If be any point outside the hyperbola then a pair of tangents PQ, PR can be drawn to it from P. The equation of pair of tangents PQ and PR is where, , , Director circle : The director circle is the locus of points from which perpendicular tangents are drawn to the given hyperbola. The equation of the director circle of the hyperbola is Example: 18 The locus of the point of intersection of tangents to the hyperbola which meet at a constant angle , is (a) (b) (c) (d) None of these Solution: (a) Let the point of intersection of tangents be . Then the equation of pair of tangents from to the given hyperbola is ......(i) From or .....(ii) Since angle between the tangents is .  . Hence locus of is . 5.3.10 Equations of Normal in Different forms . (1) Point form : The equation of normal to the hyperbola at is . (2) Parametric form: The equation of normal at to the hyperbola is = (3) Slope form: The equation of the normal to the hyperbola in terms of the slope m of the normal is (4) Condition for normality : If is the normal of then or , which is condition of normality. (5) Points of contact : Co-ordinates of points of contact are Note :  If the line will be normal to the hyperbola ,then . Important Tip  In general, four normals can be drawn to a hyperbola from any point and if be the eccentric angles of these four co-normal points, then is an odd multiple of .  If are the eccentric angles of three points on the hyperbola. , the normals at which are concurrent, then,  If the normal at P meets the transverse axis in G, then . Also the tangent and normal bisect the angle between the focal distances of P.  The feet of the normals to from lie on . Example: 19 The equation of the normal to the hyperbola at the point is (a) (b) (c) (d) Solution: (d) From Here , and  i.e., . Example: 20 If the normal at on the hyperbola meets transverse axis at G, then (Where A and are the vertices of the hyperbola) (a) (b) (c) (d) None of these Solution: (a) The equation of normal at to the given hyperbola is This meets the transverse axis i.e., x-axis at G. So the co-ordinates of G are and the co-ordinates of the vertices A and are and respectively. = = = Example: 21 The normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axis at L and M respectively, then the locus of the middle point of LM is a hyperbola whose eccentricity is (a) (b) (c) (d) None of these Solution: (a) The equation of the normal at to the hyperbola is It meets the transverse and conjugate axes at L and M, then ; Let the middle point of LM is ; then  .....(i) and  ......(ii) ; , Locus of is It is a hyperbola, let its eccentricity ; . 5.3.11 Equation of Chord of Contact of Tangents drawn from a Point to a Hyperbola . Let PQ and PR be tangents to the hyperbola drawn from any external point . Then equation of chord of contact QR is or or (At 5.3.12 Equation of the Chord of the Hyperbola whose Mid point (x1, y1) is given h Equation of the chord of the hyperbola , bisected at the given point is = i.e., Note :  The length of chord cut off by hyperbola from the line is 5.3.13 Equation of the Chord joining Two points on the Hyperbola . The equation of the chord joining the points and is Note :  If the chord joining two points and passes through the focus of the hyperbola , then . Example: 22 The equation of the chord of contact of tangents drawn from a point (2, –1) to the hyperbola is (a) (b) (c) (d) Solution: (a) From i.e., . Here, i.e., So, the equation of chord of contact of tangents drawn from a point (2, –1) to the hyperbola is i.e., Example: 23 The point of intersection of tangents drawn to the hyperbola at the points where it is intersected by the line is (a) (b) (c) (d) Solution: (a) Let be the required point. Then the equation of the chord of contact of tangents drawn from to the given hyperbola is ......(i) The given line is .....(ii) Equation (i) and (ii) represent the same line  ; Hence the required point is . Example: 24 What will be equation of that chord of hyperbola , whose mid point is (5, 3) (a) (b) (c) (d) Solution: (b) According to question, Equation of required chord is ......(i) Here = and , where ,  So, from (i) required chord is  . Example: 25 The locus of the mid-points of the chords of the circle which are tangent to the hyperbola is (a) (b) (c) (d) None of these Solution: (a) The given hyperbola is ……(i) Any tangent to (i) is ……(ii) Let be the mid point of the chord of the circle Then equation of the chord is i.e., ……(iii) Since (ii) and (iii) represent the same line.  and  = Locus of is . 5.3.14 Pole and Polar g Let P be any point inside or outside the hyperbola. If any straight line drawn through interesects the hyperbola at A and B. Then the locus of the point of intersection of the tangents to the hyperbola at A and B is called the polar of the given point with respect to the hyperbola and the point is called the pole of the polar. The equation of the required polar with as its pole is Note :  Polar of the focus is the directrix.  Any tangent is the polar of its point of contact. (1) Pole of a given line : The pole of a given line with respect to the hyperbola is (2) Properties of pole and polar (i) If the polar of passes through , then the polar of goes through and such points are said to be conjugate points. (ii) If the pole of a line lies on the another line then the pole of the second line will lie on the first and such lines are said to be conjugate lines. (iii) Pole of a given line is same as point of intersection of tangents as its extremities. Important Tips  If the polars of and with respect to the hyperbola are at right angles, then Example: 26 If the polar of a point w.r.t. touches the hyperbola , then the locus of the point is (a) Given hyperbola (b) Ellipse (c) Circle (d) None of these Solution: (a) Let be the given point. Its polar w.r.t. is i.e., This touches if    Locus of is . Which is the same hyperbola. Example: 27 The locus of the poles of the chords of the hyperbola , which subtend a right angle at the centre is (a) (b) (c) (d) Solution: (a) Let be the pole w.r.t. ......(i) Then equation of polar is .....(ii) The equation of lines joining the origin to the points of intersection of (i) and (ii) is obtained by making homogeneous (i) with the help of (ii), then  Since the lines are perpendicular, then coefficient of coefficient of or . Hence required locus is 5.3.15 Diameter of the Hyperbola . The locus of the middle points of a system of parallel chords of a hyperbola is called a diameter and the point where the diameter intersects the hyperbola is called the vertex of the diameter. Let a system of parallel chords to for different chords then the equation of diameter of the hyperbola is , which is passing through (0, 0) Conjugate diameter : Two diameters are said to be conjugate when each bisects all chords parallel to the others. If , be conjugate diameters, then . Note :  If a pair of diameters be conjugate with respect to a hyperbola, they are conjugate with respect to its conjugate hyperbola also.  In a pair of conjugate diameters of a hyperbola. Only one meets the curve in real points.  The condition for the lines to be conjugate diameters of is . Important Tips  If is the conjugate diameter of a diameter CP of the hyperbola , where P is then coordinates of D is , where C is (0, 0). Example: 28 If a pair of conjugate diameters meet the hyperbola and its conjugate in P and D respectively, then (a) (b) (c) (d) None of these Solution: (b) Coordinates of P and D are and respectively. Then = = = = . Example: 29 If the line passes through the extremities of a pair of conjugate diameters of the hyperbola then (a) (b) (c) (d) None of these Solution: (a) The extremities of a pair of conjugate diameters of are and respectively. According to the question, since extremities of a pair of conjugate diameters lie on  ……(i) Then from (i), or ……(ii) And from (ii), or ……(iii) Then subtracting (ii) from (iii) or . 5.3.16 Subtangent and Subnormal of the Hyperbolaa Let the tangent and normal at meet the x-axis at A and B respectively. Length of subtangent Length of subnormal = 5.3.17 Reflection property of the Hyperbola a If an incoming light ray passing through one focus (S) strike convex side of the hyperbola then it will get reflected towards other focus Example: 30 A ray emanating from the point (5, 0) is incident on the hyperbola at the point P with abscissa 8; then the equation of reflected ray after first reflection is (Point P lies in first quadrant) (a) (b) (c) (d) None of these Solution: (a) Given hyperbola is . This equation can be rewritten as .....(i) Since x coordinate of P is 8. Let y-coordinate of P is lies on (i) ; P lies in first quadrant) Hence coordinate of point P is Equation of reflected ray passing through and );  Its equation is or or 5.3.18 Asymptotes of a Hyperbola a An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity. The equations of two asymptotes of the hyperbola are or . Note :  The combined equation of the asymptotes of the hyperbola is .  When i.e. the asymptotes of rectangular hyperbola are , which are at right angles.  A hyperbola and its conjugate hyperbola have the same asymptotes.  The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only i.e. Hyperbola – Asymptotes = Asymptotes – Conjugated hyperbola or, .  The asymptotes pass through the centre of the hyperbola.  The bisectors of the angles between the asymptotes are the coordinate axes.  The angle between the asymptotes of the hyperbola i.e., is or 2 .  Asymptotes are equally inclined to the axes of the hyperbola. Important Tips  The parallelogram formed by the tangents at the extremities of conjugate diameters of a hyperbola has its vertices lying on the asymptotes and is of constant area. Area of parallelogram 4(Area of parallelogram QDCP) = Constant  The product of length of perpendiculars drawn from any point on the hyperbola to the asymptotes is . Example: 31 From any point on the hyperbola, tangents are drawn to the hyperbola . The area cut-off by the chord of contact on the asymptotes is equal to (a) (b) (c) (d) Solution: (d) Let be a point on the hyperbola , then The chord of contact of tangent from P to the hyperbola is .....(i) The equation of asymptotes are = 0 ......(ii) And = 0 ......(iii) The point of intersection of the asymptotes and chord are , (0, 0) Area of triangle = = . Example: 32 The combined equation of the asymptotes of the hyperbola [Karnataka CET 2002] (a) (b) (c) (d) Solution: (d) Given, equation of hyperbola and equation of asymptotes ......(i) which is the equation of a pair of straight lines. We know that the standard equation of a pair of straight lines is Comparing equation (i) with standard equation, we get , and . We also know that the condition for a pair of straight lines is . Therefore, or or Substituting value of  in equation (i), we get . 5.3.19 Rectangular or Equilateral Hyperbola a (1) Definition : A hyperbola whose asymptotes are at right angles to each other is called a rectangular hyperbola. The eccentricity of rectangular hyperbola is always . The general equation of second degree represents a rectangular hyperbola if   0, and coefficient of + coefficient of = 0 The equation of the asymptotes of the hyperbola are given by . The angle between these two asymptotes is given by . If the asymptotes are at right angles, then . Thus the transverse and conjugate axis of a rectangular hyperbola are equal and the equation is . The equations of the asymptotes of the rectangular hyperbola are and . Clearly, each of these two asymptotes is inclined at to the transverse axis. (2) Equation of the rectangular hyperbola referred to its asymptotes as the axes of coordinates : Referred to the transverse and conjugate axis as the axes of coordinates, the equation of the rectangular hyperbola is …..(i) The asymptotes of (i) are y = x and y = – x. Each of these two asymptotes is inclined at an angle of with the transverse axis, So, if we rotate the coordinate axes through an angle of keeping the origin fixed, then the axes coincide with the asymptotes of the hyperbola and and . Substituting the values of x and y in (i), We obtain the where . This is transformed equation of the rectangular hyperbola (i). (3) Parametric co-ordinates of a point on the hyperbola XY = c2 : If t is non–zero variable, the coordinates of any point on the rectangular hyperbola can be written as . The point on the hyperbola is generally referred as the point ‘t’. For rectangular hyperbola the coordinates of foci are and directrices are . For rectangular hyperbola , the coordinates of foci are and directrices are . (4) Equation of the chord joining points t1 and t2 : The equation of the chord joining two points on the hyperbola is . (5) Equation of tangent in different forms (i) Point form : The equation of tangent at to the hyperbola is or (ii) Parametric form : The equation of the tangent at to the hyperbola is .On replacing by and by on the equation of the tangent at i.e. we get . Note :  Point of intersection of tangents at and is (6) Equation of the normal in different forms : (i) Point form : The equation of the normal at to the hyperbola is . As discussed in the equation of the tangent, we have So, the equation of the normal at is This is the required equation of the normal at . (ii) Parametric form: The equation of the normal at to the hyperbola is . On replacing by and by in the equation. We obtain Note :  The equation of the normal at is a fourth degree in t. So, in general, four normals can be drawn from a point to the hyperbola  If the normal at on the curve meets the curve again in ' ' then; .  Point of intersection of normals at and is Important Tips  A triangle has its vertices on a rectangular hyperbola; then the orthocentre of the triangle also lies on the same hyperbola.  All conics passing through the intersection of two rectangular hyperbolas are themselves rectangular hyperbolas.  An infinite number of triangles can be inscribed in the rectangular hyperbola whose all sides touch the parabola . Example: 33 If represents a rectangular hyperbola, then equals (a) 5 (b) 4 (c) – 5 (d) None of these Solution: (c) Since the general equation of second degree represents a rectagular hyperbola if and coefficient of coefficient of . Therefore the given equation represents a rectangular hyperbola if i.e., Example: 34 If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus, the mid-point of PN is (a) Circle (b) Parabola (c) Ellipse (d) Hyperbola Solution: (d) Let be the rectangular hyperbola, and let be a point on it. Let be the mid-point of PN. Then the coordinates of Q are . and  and But lies on xy = c2.  Hence, the locus of is , which is a hyperbola. Example: 35 If the normal at on the curve meets the curve again in t, then (a) (b) (c) (d) Solution: (a) The equation of the tangent at is If it passes through then     Example: 36 If the tangent and normal to a rectangular hyperbola cut off intercepts and on one axis and and on the other axis, then (a) (b) (c) (d) None of these Solution: (c) Let the hyperbola be . Tangent at any point t is Putting and then intercepts on the axes are and Normal is . Intercepts as above are , = ; . Example: 37 A variable straight line of slope 4 intersects the hyperbola at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is [IIT 1997] (a) (b) (c) (d) None of these Solution: (a) Let be any point on the locus. Equation of the line through P and having slope 4 is .....(i) Suppose this meets ......(ii) in and Eliminating y between (i) and (ii), we get   ......(iii) This has two roots say ; ......(iv) and ......(v) Also, [ P divides AB in the ratio 1 : 2] i.e., ......(vi) (vi) – (iv) gives, and Putting in (v), we get   Required locus of is . Example: 38 PQ and RS are two perpendicular chords of the rectangular hyperbola . If C is the centre of the rectangular hyperbola, then the product of the slopes of CP, CQ, CR and CS is equal to (a) – 1 (b) 1 (c) 0 (d) None of these Solution: (b) Let be the parameters of the points P, Q, R and S respectively. Then, the coordinates of P, Q, R and S are , , and respectively. Now,    .....(i) Product of the slopes of and [Using (i)] 5.3.20 Intersection of a Circle and a Rectangular Hyperbola a If a circle cuts a rectangular hyperbola in A, B, C and D and the parameters of these four points be and respectively; then (1) (i) (ii) (iii) (iv) (v) (2) Orthocentre of is but D is Hence H and D are the extremities of a diagonal of rectangular hyperbola. (3) Centre of mean position of four points is i.e., Centres of the circles and rectangular hyperbola are (– g, – f) and (0, 0); mid point of centres of circle and hyperbola is . Hence the centre of the mean position of the four points bisects the distance between the centres of the two curves (circle and rectangular hyperbola) (4) If the circle passing through ABC meet the hyperbola in fourth points D; then centre of circle is (–g, –f) i.e., Example: 39 If a circle cuts a rectangular hyperbola in A, B, C, D and the parameters of these four points be and respectively. Then [Kurukshetra CEE 1998] (a) (b) (c) (d) Solution: (b) Let the equation of circle be ......(i) Parametric equation of rectangular hyperbola is Put the values of x and y in equation (i) we get  Hence product of roots Example: 40 If the circle intersects the hyperbola in four points , then (a) (b) (c) (d) Solution: (a,b,c,d) Given, circle is .......(i) and hyperbola be .....(ii) from (ii) . Putting in (i), we get  , Since both the curves are symmetric in x and y, ; .